A Class of Exponential Integrators Based on Spectral Deferred Correction

TL;DR

This paper introduces a new class of high-order exponential integrators called ETDSDC, which outperform existing methods in accuracy, efficiency, and robustness for solving PDEs, especially when combined with spectral spatial methods.

Contribution

The paper presents a novel class of exponential integrators based on spectral deferred correction, demonstrating improved accuracy, stability, and immunity to order reduction over existing schemes.

Findings

ETDSDC methods have larger accuracy regions.

ETDSDC schemes are more efficient for high-accuracy PDE solutions.

ETDSDC schemes are immune to severe order reduction.

Abstract

We introduce a new class of arbitrary-order exponential time differencing methods based on spectral deferred correction (ETDSDC) and describe a simple procedure for initializing the requisite matrix functions. We compare the stability and accuracy properties of our ETDSDC meth- ods to those of an existing implicit-explicit spectral deferred correction scheme (IMEXSDC). We find that ETDSDC methods have larger accuracy regions and comparable stability regions. We conduct numerical experiments to compare ETD and IMEX spectral deferred correction schemes against a competing fourth-order ETD Runge-Kutta scheme. We find that high-order ETDSDC schemes are the most efficient in terms of function evaluations and overall speed when solving partial differential equations to high accuracy. Our results suggest that high-order ETDSDC schemes are well-suited to work in conjunction with spectral spatial methods or other high-order spatial discritizations. Addi- tionally, ETDSDC schemes appear to be immune to severe order reduction, a problem which affects other ETD and IMEX schemes, including IMEXSDC.